Browse › Algebraic and complex geometry › 058 Semialgebraic universal covers and bounded domains
Symmetry of semialgebraic bounded domains with compact quotient
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Abstract
Every nonempty connected semialgebraic bounded open subset of a complex affine variety admitting a properly discontinuous cocompact group of biholomorphisms is smooth and biholomorphic to a bounded symmetric domain. This answers the bounded-domain question of Kollár and Pardon affirmatively.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope note 058: Semialgebraic universal covers and bounded domains (lean/docs/058.md, 2026). The scope note reports formalization of this paper's main theorem (smoothness and biholomorphism to a bounded symmetric domain under proper cocompact action), allowing finite stabilizers and a singular affine ambient variety, and links the comparator statement.
- OpenAI math release, SymmetricDomains.lean comparator statement (lean/ComparatorChallenges, 2026). Lean statement of the main theorem (bounded, connected, semialgebraic, relatively open U in an affine algebraic set; proper holomorphic action of a discrete group with compact quotient implies U is smooth and biholomorphic to a bounded symmetric domain), with the proof in this comparator file left as
sorry.
Related prior literature mentioned by others
- János Kollár and John Pardon, Algebraic varieties with semialgebraic universal cover (arXiv:1104.2309v2, 2011-12-15; Journal of Topology, 2012). The AI Proved What? family exposition identifies this as the source of both the product-classification conjecture (Conjecture 2) and the bounded-domain question addressed by the two manuscripts. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Le Kuai, Kefeng Liu, Yunhui Wu, Rigidity of complex bounded domains with finite-volume quotients (arXiv:1801.00459, v1 2018, v3 2 Oct 2026). Proves that bounded convex domains with finite-volume quotients are bounded symmetric and addresses Kazhdan's conjecture, which sits next to the paper's discussion of Frankel's convex compact-quotient theorem.
- Alan M. Nadel, Semisimplicity of the group of biholomorphisms of the universal covering of a compact complex manifold with ample canonical bundle (Ann. of Math. 132 (1990), 193–211). Concerns automorphism groups of domains whose compact quotients have ample canonical bundle, which is the situation the introduction recalls from Kollár–Pardon §3.
- A. V. Isaev and S. G. Krantz, Domains with non-compact automorphism group: a survey (arXiv:math/9612201, 1997). Surveys the Wong–Rosay ball characterisation and boundary orbit-accumulation methods, which the paper's first step extends to singular affine ambient spaces.
- Andrew M. Zimmer, Rigidity of complex convex divisible sets (arXiv:1308.4116, 2013). An earlier rigidity theorem for complex convex sets with cocompact automorphism groups; the paper itself cites only Zimmer's later C^{1,1}-boundary result.
- Vasily Rogov, On the metric Kollár-Pardon problem (arXiv:2401.07983, 2024-01-15). Its homogeneity theorem for aspherical universal covers with definable lifted metrics offers a comparison for the paper's passage from semialgebraicity and compact quotient to symmetry, under additional metric hypotheses.
Digests and expositions
- AI Proved What?, Three building blocks? Semialgebraic universal covers and bounded domains (episode 058, October 2026). This video and transcript outline both arguments, identify the classification's dependence on other release manuscripts, and distinguish the bounded-domain Lean result from the broader classification. [AI-generated per source] [family-level]
Elsewhere
- Emergent Mind: family 058
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026,
author = {{OpenAI}},
title = {{Symmetry of semialgebraic bounded domains with compact quotient}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026/paper.pdf}{OAI:Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/058.md. The Lean statements have not been checked by people against the paper's claims.
Family 058: Semialgebraic universal covers and bounded domains
Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products $D\times\mathbb C^m\times F$, with D bounded symmetric and F simply connected, normal, and projective. A universal cover is quasi-projective exactly when the bounded symmetric factor is absent. In particular, a smooth projective variety covered by ℂn has a finite étale cover by an abelian variety.
Other papers in this family:
- Semialgebraic universal covers of normal projective varieties · September 24, 2026